AcknowledgmentsIthankProfFrancoisLabourieforhisguidanceIamalsothankfultotheAlgantConsortiumforgivingmetheopportunitytostudyintheAlgantMastersProgrammeandsupportingmebytheirscholarshipIamgratefultoM ID: 333351
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MargulisSpaceTime:CrookedPlanesandTheMargulisInvariantSouravGhoshAdvisor:FrancoisLabourieMasterThesis2012 AcknowledgmentsIthankProf.FrancoisLabourieforhisguidance.IamalsothankfultotheAlgantConsortiumforgivingmetheopportunitytostudyintheAlgantMastersProgrammeandsupportingmebytheirscholarship.IamgratefultoMr.ChinmoyChowdhuryandDr.MahanMjforintroducingmetotheworldofMathematics.Finally,IwouldliketoexpressmygratitudetowardsmyparentsMr.SwapanGhoshandMrs.PurnimaGhoshandalsotowardsSaumyaShuklafortheirconstantemotionalsupport.ii Contents1Introduction22GeometryofR2;132.1TheHyperboloidModel............................32.2TheLorentzProduct..............................43AneGeometry63.1AneSpacesandtheirautomorphisms....................63.2Anedeformationsoflinearactions.....................73.3TheMargulisInvariant.............................74MargulisSpaceTimes104.1FundamentalPolyhedraforthelinearpart..................104.2AneFundamentalPolyhedra........................115TheGeometryofCrookedPlanes155.1AnatomyofaCrookedPlane.........................155.2IntersectionofWings..............................165.3IntersectionofStemwithWing........................175.4IntersectionofStems..............................195.5IntersectionofCrookedPlanes........................196TheExtendedMargulisInvariantandItsApplications226.1FlatBundlesassociatedtoAneDeformations...............226.2Labourie'sdi usionoftheMargulisInvariant................236.3AnequivalentconditionforProperness....................24Bibliography261 Chapter1IntroductionIn1977,Milnor[2]askedwhetheranon-amenablegroup(e.gafreegroupofrank2)couldactproperlybyanetransformations.HeobservedthatbyapplyingTitsalternative[3],thisquestionisequivalenttowhetheranon-abelianfreegroupcouldactproperlybyanetransformations.Furthermore,heproposedthefollowingconstructionofsuchagroup:StartwithafreediscretesubgroupofSO0(2;1)(forexampleaSchottkygroupactingonthehyperbolicplane)andaddtranslationcomponentstoobtainagroupofanetransformationswhichmayactfreely.In1983,MargulisintroducedtheMargulisinvariantandusedittoshowthatsuchproperlydiscontinuousgroupsdoexist,realizingMilnor'ssuggestion.In1991,Drumm[5]gaveageneralizationofSchottky'sconstructionintroducingtheCrookedPlanes.FollowingthisworkDrummandGoldman[6][9]inaseriesofpapersstudiedtheCrookedPlanesinmuchdetail.ThisapproachgaveaslightlystrongerpositiveresultastowhichgroupsactproperlydiscontinuouslyonRn.Intheabovementioned1983paperMargulis[4]usedtheMargulisinvarianttodetectproperness.Hegaveanecessaryconditionforpropernessusingthesignoftheinvariant.Goldmanconjecturedthisnecessaryconditiontobesucient.In2006Charette[12]cameupwithanexampleofaone-holedtorussuggestingthattheconjectureisgenerallyfalse.In2001,followingMargulis'swork,Labourie[10]extendedtheoriginalMargulisinvarianttohigherdimensions.Recently,in2009,Margulis,GoldmanandLabourie[7]foundanequivalentconditionforpropernessusingtheextendedMargulisinvariantandgaveaveryconceptualproofoftheOppositeSignLemma.InthispaperwestudytheworksofDrummandGoldmanonCrookedPlanes[5][6][9]andtherecentworksofMargulis,GoldmanandLabourie[7]ongivinganequivalentconditionforproperness.2 Chapter2GeometryofR2;1Let,R2;1bethe3-dimensionalrealvectorspacewithinnerproduct,B(v;w):=v1w1+v2w2�v3w3andG0=O0(2;1)theidentitycomponentofitsgroupofisometries.G0consistsoflinearisometriesofR2;1whichpreservebothanorientationofR2;1andaconnectedcomponentoftheopenlightcone:fv2R2;1:B(v;v)0g.Then,G0=O0(2;1)=PSL(2;R)=Isom0(H2),whereH2denotestherealhyperbolicplane.2.1TheHyperboloidModelWedeneH2asfollows.WeworkintheirreduciblerepresentationR2;1(isomorphictotheadjointrepresentationofG0).Anonzerovectorv2R2;1iscalled:SpacelikeifB(v;v)&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;0LightlikeorNullifB(v;v)=0TimelikeifB(v;v)0Thetwosheetedhyperboloid:fv2R2;1:B(v;v)=�1ghastwoconnectedcomponents.Wexatimelikevectore3anddeneH2asthecon-nectedcomponentwithB(v;e3)0.TheLorentzianmetricdenedbyBrestrictstoaRiemannianmetricofconstantcurvature�1onH2.TheidentitycomponentG0oftheisometrygroupofR2;1isthegroupoforientationpreservingisometriesofH2.3 2.2TheLorentzProductOnthespaceR2;1denedasabovewedeneacrossproductcalledtheLorentziancrossproductasfollows::R2;1R2;1�!R2;1suchthat,(u;v)7�!24u2v3�u3v2u3v1�u1v3u2v1�u1v235.Notethatisaskew-symmetricandbilinearform,satisfyingthefollowingproperties:B(u;uv)=B(v;uv)=0uv=�vuB(uv;uv)=B(v;u)2�B(u;u)B(v;v)Letk:kbetheeuclideannormonthespaceR3.Denotebyv1;v2;:::;vn&#x-278;thesubspacegeneratedbyv1;v2;:::;vnandthesphereofuniteuclideanlengthbyS2.Foranyv2R2;1itsLorentzperpendicularplaneisdenotedbyP(v).LetCbethelightcone.ItconsistsofallnullvectorsanditstwocomponentsformthetimeorientationsofR2;1.WedeneC+andC�respectivelytobethepositiveandthenegativetimeorientation.SimilarlywedenotebyUthespaceofalltimelikevectorsanditstwocomponentsasU+andU�.Ifa;b2C+\S2aretwononzeronullvectorswithpositivetimeorientation,thenv=abisspacelikeandisavectorparalleltotheintersectionP(a)\P(b).LetvbeanullvectorthentheplaneP(v)equalstheplanetangenttothelightconecontainingthelineRvandthenulllineRvdividestheplaneintotwocomponents.Themappingx7�!xvdenesadi eomorphismofC+nR+vontoonecomponent.WedenotetheclosureofthiscomponentbyP+(v),thatis,P+(v):=cl((C+nR+v)v)andtheothercomponentbyP�(v),thatis,P�(v):=cl(v(C+nR+v)).IfvisaspacelikevectorthenP(v)\Cistheunionoftwonulllines.Thereexistsauniquepairx�(v);x+(v)2P(v)\C+suchthatkx�(v)k=kx+(v)k=1andB(v;x�(v)x+(v))&#x-278;0.Ifvisaspacelikethen,x�(v)x+(v)=2B(v;v)v=(B(v;v)+kvk2)vx+(v)=�B(v;v)1=2x+(v).4 WedeneaconicalneighbourhoodAC+ofv2C+tobeanopenconnectedsubsetofC+containingvsuchthatifw2AthenR+wA.FordisjointconicalopensetsAandBwedene,T(A;B):=fv2R2;1:B(v;ab)�0foralla2cl(A)andb2cl(B)g:Inparticular,ifA,BareconnectedthenT(A;B)isanopeninnitepyramidwhosevertexistheoriginandwhosefouredgesareparalleltovectorsintheboundaryofAandB.5 Chapter3AneGeometryInthischapterwecollectgeneralpropertiesofanespaces,anetransformationsandanedeformationsoflineargroupactions.Weareprimarilyinterestedinanede-formationsoflinearactionsfactoringthroughtheirreducible2r+1dimensionalrealrepresentationVrofG0whererisapositiveinteger.Wenotethatforr=1wegetV1=R2;1.3.1AneSpacesandtheirautomorphismsLetVbearealvectorspace.AnanespaceE(modelledonV)isaspaceequippedwithasimplytransitiveactionofV.WecallVthevectorspaceunderlyingE,andrefertoitselementsastranslations.Translation vbyavectorv2Visdenotedbyaddition,thatis, v:E�!Egivenbyx7�!x+v.LetEbeananespacewithassociatedvectorspaceV.ChoosinganarbitrarypointO2E(theorigin)identiesEwithVviathemap,f:V�!Egivenbyv7�!O+v.AnaneautomorphismofEisthecompositionofalinearmapping(usingtheaboveidenticationofEandV)andatranslation,thatis,g:E�!EgivenbyO+v7�!O+L(g)(v)+u(g)whereL(g)2GL(V)andu(g)2V.TheaneautomorphismsofEformagroupA (E).Themapping,(L;u):A (E)�!GL(V)nVgivenbyg7�!(L(g);u(g))givesanisomorphismofgroups.ThelinearmappingL(g)2GL(V)iscalledthelinearpartoftheanetransformationg,andL:A (E)�!GL(V)givenbyg7�!L(g)isahomomorphism.Thevectoru(g)2Viscalledthetranslationalpartofg.Themapping,6 u:A (E)�!Vgivenbyg7�!u(g)satisesthefollowingidentity(alsoknownasthecocycleidentity):u(g1g2)=u(g1)+L(g1)u(g2)forg1;g22A (E).3.2AnedeformationsoflinearactionsLet�0GL(V)beagroupoflinearautomorphismsofavectorspaceV.Denotethecorresponding�0moduleasVaswell.Ananedeformationof�0isarepresentation,:�0�!A (E)suchthatListheinclusion�0,!GL(V).Weconfusewithitsimage�:=(�0),towhichweasorefertoasananedeformationof�0.Notethatembedds�0asthesubgroup�ofGL(V).Intermsofthesemi-directproductdecompositionA (E)=GL(V)nVananedeformationisthegraph=u(withimagedenotedby�=�u)ofacocycleu:�0�!V,thatis,amapsatisfyingtheaforementionedcocycleidentity.Wewriteg=(g0)=(g0;u(g0))2�0nVforthecorrespondinganetransformationg(x)=g0(x)+u(g0).CocyclesformavectorspaceZ1(�0;V).Cocyclesu1;u22Z1(�0;V)arecohomologousiftheirdi erenceu1�u2isacoboundary,acocycleoftheform,v0:�0�!Vgivenbyg7�!v0�gv0wherev02V.Moreover,cohomologousclassesofcocyclesformavectorspaceH1(�0;V).Anedeformationsu1;u2areconjugatebytranslationbyv0ifandonlyifu1�u2=v0.ThusH1(�0;V)parametrizestranslationalconjugacyclassesofanedeformationsof�0GL(V).Notethatwhenu=0,theanedeformation�uequals�0itself.3.3TheMargulisInvariantConsiderthecasethatG0=PSL(2;R)andLisanirreduciblerepresentationofG0.Foreverypositiveintegerr,letLrdenotetheirreduciblerepresentationofG0onthe2r-symmetricpowerVrofthestandardrepresentationofSL(2;R)onR2.ThedimensionofVrequals2r+1.Thecentralelement�I2SL(2;R)actsby(�1)2r=1.SothisrepresentationofSL(2;R)denesarepresentationofPSL(2;R)=SL(2;R)=fIg.NotethattherepresentationR2;1introducedbeforeisV1,thecasewhenr=1.Fur-thermoretheG0invariantnon-degenerateskew-symmetricbilinearformonR2inducesanon-degeneratesymmetricbilinearformBonVr,whichwenormalizeinthefollowingparagraph.7 Anelementg2G0ishyperbolicifitcorrespondstoanelement~gofSL(2;R)withdis-tinctrealeigenvalues.Necessarilytheseeigenvaluesarereciprocals;�1whichwecanuniquelyspecifybyrequiringjj1.Furthermorewechooseeigenvectorsv+;v�2R2suchthat:~g(v+)=v+~g(v�)=�1v�Theorderedbasisfv�;v+gispositivelyoriented.ThentheactionLrhaseigenvalues2j,forj2f�r;1�r;:::;0;:::;r�1;rgwherethesymmetricproductvr�j�vr+j+2Vrisaneigenvectorwitheigenvalue2j.Inparticulargxesthevectorx0(g):=cvr�vr+,wherethescalarcischoosensothatB(x0(g);x0(g))=1.Callx0(g)theneutralvectorofg.Thesubspaces,V�(g):=rXj=1R(vr+j�vr�j+),V+(g):=rXj=1R(vr�j�vr+j+)areginvariantandVenjoysag-invariantB-orthogonaldirectsumdecomposition,V=V�(g)R(x0(g))V+(g).Foranynormk:konV,thereexistsC;k&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.45; 0 ;&#xTd [;0suchthat,kgn(v)k6Ce�knkvkforv2V+(g)andkg�n(v)k6Ce�knkvkforv2V�(g).Furthermore,x0(gn)=jnjx0(g)ifn2Znf0gandV(gn)=V(g)ifn&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.45; 0 ;&#xTd [;0V(g)ifn0.Now,letussupposethatg2A (E)isananetransformationwhoselinearpartL(g)ishyperbolic.ThenthereexistsauniqueanelinelgEwhichisg-invariant.Thelinelgisparalleltox0(L(g)).Therestrictionofgtolgisatranslationbythevector,B(gx�x;x0(L(g)))x0(L(g))wherex0(L(g))isasdenedabovewithB(x0(L(g));x0(L(g)))=1.Supposethat�0G0beaSchottkygroup,thatis,anon-abeliandiscretesubgroupcontainingonlyhyperbolicelements.Suchadiscretesubgroupisafreegroupofrankatleasttwo.WedenetheMargulisinvarianttobethefunction, u:�0�!Rgivenbyg7�!B(u(g);x0(g))whereu2Z1(�0;V).TheMargulisinvariant uassociatedtoananedeformation�uisawelldenedclassfunctionon�0satisfyingthefollowingproperties:8 [u](gn)=jnj [u](g)forg2�0and[u]2H1(�0;V) [u](g)=0,gxesapointinV.Thefunction [u]dependslinearlyon[u].Themap :H1(�0;V)�!Rgivenby[u]7�! [u]isinjective.Inhis1983paperMargulisusedthisinvarianttodetectproperness.ThesignicanceoftheMargulisinvariantcomesfromthefollowingresultduetoMargulis:Theorem3.3.1(TheOppositeSignLemma).Suppose�actsproperly.Theneither [u](g)�0forallg2�0or [u](g)0forallg2�0.WewillgiveaproofofthisresultatthelastchapterusingtoolsdevelopedjointlybyMargulis,GoldmanandLabourie.9 Chapter4MargulisSpaceTimesCompleteanely atmanifoldscorrespondtosubgroups�A (Rn)whichactproperlydiscontinuouslyonRn,and1(M)=�forM=Rn=�.Milnorshowedthatif�isvirtuallypolycyclicthenthereexistssomecompleteanely atmanifoldMsuchthat1(M)=�,andheaskediftheconversewastrue.Margulisdemonstratedthatthereexistfreesubgroups�A (R3)actingproperlydis-continuouslyonR3,thusansweringMilnor'squestionnegatively.ByFriedandGoldman,theunderlyinglineargroupof�mustbeconjugatetoasubgroupofG0.Thecorrespond-ingquotientmanifoldsM=R3=�arecalledMargulisspace-times.Wenotethat�,agroupofanehomeomorphismsofR3actsproperlydiscontinuouslyandfreelyonR3ifthereexistsathreedimensionalsubmanifoldXwithboundary(afundamentaldomain)suchthatnotwoelementsoftheinteriorofXare�-equivalentandeveryelementofR3is�-equivalenttoanelementofX.InthefollowingsectionswetrytoconstructanicefundamentaldomainforalargeclassofMargulisspace-times.4.1FundamentalPolyhedraforthelinearpartLetG=g1;g2;:::;gn&#x-278;wheregi2G0fori2f1;2;:::;ng.DenoteGi=gi&#x-278;.Wenotethat,G\actsasaSchottkygrouponC+"[9]ifthereareconicalneighbourhoodsAiofx(gi)suchthatcl(Ai)\cl(Ai[i6=j(A+j[A�j)=;andcl(gi(A�i))=C+nA+i.DenotethesetCnfv2C:v=kx(gi)forsomek2RgbyCandthecomplementoftheset(A+i[(�A+i)[A�i[(�A�i))byA.Theorem4.1.1.AisafundamentaldomainfortheactionofGionC.Proof.Wenotethatcl(gi(A�i))=C+nA+i.Usingthiswegetthatforg2Giandg6=ewehavegi(A)\A=;.Now,ifwetakex2Cthengni(x)!x+(gi).Soforanyy2Cthereexistsax2Asuchthatgmi(y)=xforsomeintegerm.HenceAisafundamentaldomainfortheactionofGionC. 10 Denevijforj2f1;2gtobeofeuclideannorm1andintheboundaryofAisatisfyingvi1vi26=0andgi(v�ij)=kgi(v�ij)k=v+ij.Theactionunderconsiderationislinear.SousingtheabovetheoremandlinearityoftheactionwegetthatafundamentaldomainfortheactionofGionU+istheregionboundedbyA,F+i:=fv2v+i1;v+i2&#x-278;:B(v;v)60gandF�i:=fv2v�i1;v�i2&#x-278;:B(v;v)60g.ConsiderthehalfplanesP(vij).Notethatgi(P+(v�ij))=P+(v+ij)andgi(P�(v�ij))=P�(v+ij),sinceg(xv)=kg(xv)k=xg(v)forhyperbolicg2G0andv2R2;1.DenethewedgesWitobetheopenregionboundedbyFi[P+(vi1)[P+(vi2)andnotcontainingFi[P+(vi1)[P+(vi2)andthewedgesMitobetheopenregionboundedbyFi[P�(vi1)[P�(vi2)andnotcontainingFi[P�(vi1)[P�(vi2).Theorem4.1.2.AfundamentaldomainfortheactionofGionR2;1n(P+(x�gi)[P+(x+gi))isR2;1n(W+i[W�i)andonR2;1n(P�(x�gi)[P�(x+gi))isR2;1n(M+i[M�i).Proof.Theresultfollowsfromthepreviousdiscussionandusingthelasttheorem. 4.2AneFundamentalPolyhedraInthissectionwewilldiscussaboutthefundamentaldomainsinvolvingtheP+(v)'s.TheargumentforthefundamentaldomainsforP�(v)iscompletelyanalogous.LetH=h1;h2;:::;hn&#x-278;A (R3)issuchthatL(hi)=giandu(hi)=viwherevi2R3.DenoteHi=hi&#x-278;.Thefundamentaldomainofacyclicanegroupisboundedbytranslatesofcomponentsoftheboundaryofafundamentaldomainforthecorrespondingcycliclineargroup.Wenotethatcl(hi(W�i))=cl(gi(W�i)+vi)=(R2;1nW�i)+viandafundamentaldomainfortheactionofHionR2;1isthecomplementofW�iandW+i+viifthesetwosetshavedisjointclosures.Let%denotetheeuclideandistancebetweentwopointsinR3.If%(y;z+vi)&#x-278;0forallchoicesofy2cl(W�i)andz2cl(W+i)thencl(W�i)\cl(W+i+vi)=;.Inparticular,ifforeachpairofvectorsy2cl(W�i)andz2cl(W+i)thereisavectoru2R2;1suchthatB(y;u)6=B(z+vi;u)thencl(W�i)\cl(W+i+vi)=;.Toconstructugiventhevectorsy2W�iandz2W+i,rstexaminethecaseinwhichyandzarebothspacelike.Inthiscase,x+y2cl(A�i),x+z2cl(A+i)andu=x+yx+z.11 Ifyisnotspacelikeandzisspacelike,letu=y=kykx+z.Ifzisnotspacelikeandyisspacelike,letu=x+yz=kzk.Ifneitherynorzarespacelike,letu=(y=kyk)(z=kzk).NotethatB(y;u)=0ifyisnotspacelike.Ifyisspacelikethenx+y=x�uandB(y;u)0.Similarly,B(z;u)&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;0.IfB(vi;u)&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;0forallpossiblevectorsudescribedabovethencl(W�i)\cl(W+i+vi)=;andafundamentaldomainforHiisthecomplementofW�iandW+i+vi.Fory2cl(W�i)andz2cl(W+i)onecanconstructavectoruasabovesuchthatB(z+vi;u)=B(z;u)+B(vi;u)&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;kandB(y;u)60.Thesetofvi'ssuchthatB(vi;u)&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;0foraxeduisahalfspaceboundedbyP(u).ThesetoftranslationsgivingrisetoafundamentaldomainforthegivenAiinthisconstructionisT(A�i;A+i),thesetofallowabletranslations.Anothersetofallowabletranslationsisobtainedbynotingthatcl(hi(W�i�g�1i(vi)))=R2;1nW+i.InthiscasethewedgesseparateifB(�g�1i(vi);u)0forallu=zwwherew2cl(A+i)andz2cl(A�i).Equivalently,B(vi;gi(u))&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.60; 0 ;&#xTd [;0forallgi(u)=zwwherew2cl(gi(A+i))andz2cl(gi(A�i))andthesetofallowabletranslationisT(gi(A�i);gi(A+i)).Thesetwosetsofallowabletranslationscanbecombinedtogethertomakealargerthirdsetofallowabletranslations.Ifvi12T(A�i;A+i)andvi22T(gi(A�i);gi(A+i))thenvi=vi1+vi2isalsoanallowabletranslation.Weseethatthesetfv:vi=vi1+vi2wherevi12T(A�i;A+i)andvi22T(gi(A�i);gi(A+i))gissameasthesetT(A�i;gi(A+i)).DeneW�i:=W�i�g�1i(vi2)andW+i:=W+i+vi1.Theorem4.2.1.AfundamentaldomainfortheactionofHionR2;1isR2;1n(W�i[W+i),ifvi2T(A�i;gi(A+i))wherevi12T(A�i;A+i)andvi22T(gi(A�i);gi(A+i))aresuchthatvi=vi1+vi2.Proof.Theresultfollowseasilyfromthediscussionabove. ThefundamentaldomainfortheactionofHonR2;1istheintersectionofthefunda-mentaldomainsoftheactionofHi'sonR2;1providedthefundamentaldomainforeachHicompletelycontainsthecomplementofthefundamentaldomainfortheotherHj'swherej6=i.InpassingfromthefundamentaldomainofHitothefundamentaldomainofH,itisusefultodemandbothwedgesbetranslatedawayfromtheorigin.Wenote12 that,inordertoguaranteethattheclosuresofthetranslatedwedgesaredistinct,itisusefultoconsidereachwedgepairedwiththeotherwedges.LetAi=Ai[j6=i(A+j[A�j).Ifvi2T(gi(A+i);A�i)thenvi=vi1+vi2forsomevi12T(A+i;A�i)andvi22T(gi(A+i);gi(A�i),andif�vi2T(gi(A+i);A�i)thenvi=vi1+vi2forsome�vi12T(A+i;A�i)and�vi22T(gi(A+i);gi(A�i).Inthiscase,letW+i=W+i+vi1,W�i=W�i�g�1i(vi2),M+i=M+i+vi1andM�i=M�i�g�1i(vi2).Theorem4.2.2.Lethi(x)=gi(x)+vifori2f1;2;:::;ngandH=h1;h2;:::;hn&#x-278;.IfL(H)actsasaSchottkysubgrouponC+and:vi2T(gi(A+i);A�i),thenHactsproperlydiscontinuouslyonR2;1.Inthiscase,R2;1n([i(W�i[W+i))isafundamentalpolyhedronfortheactionofHonR2;1.�vi2T(gi(A+i);A�i),thenHactsproperlydiscontinuouslyonR2;1.Inthiscase,R2;1n([i(M�i[M+i))isafundamentalpolyhedronfortheactionofHonR2;1.Proof.Itsucestoprovethetheoremforthecasevi2T(gi(A+i);A�i).ItisclearfromtheconstructionthatnotwoelementsofX:=R2;1n([i(W�i[W+i))areH-equivalent.Assumethatthereexistsap2R2;1whichisnotH-equivalenttoanypointinX.Thus,onecanconstructaninnitesequenceofembeddedimagesofthewedgesallcontainingpinthefollowingmanner:Let 0=eandWj0i0=!.Forintegersn&#x-278;1choose n2H,in2f1;2gandjn2f�1;1g,sothatp2 n(Wjnin), n+1(Wjn+1in+1) n(Wjnin)and n+1= nhjnin.Theleadingtermof nishj0i0andbyanapplicationoftheBrouwerxedpointtheorem[9]itcanbeshownthatx+( n)2Aj0i0.Let n+1=2isdenedtobehk n,wherek2f1;2gischosensothatk6=i0.UsingthesameargumentandapplyingtheBrouwerxedpointtheorem[9]onegetsthatx+( n+1=2)2gk(Aj0i0).DenetheplaneSm:=x+( m);x0( m)&#x-278;forallm2f0;1=2;1;3=2;2;:::g.ConsidertheintersectionoftheembeddedimagesofthewedgesandtheplaneP:=fx2R2;1:x3=p3g.pisH-equivalenttoelementsinallofthewedgesWi.Onecanassumethatpiscontainedina"small"wedge!,wheretheanglebetweeneverypairofrayscontainedin!\Pis6=2.Inparticular,Sn\Pcontainsaraylyingcompletelywithin!\Pforallpositiveintegersn.ChooseL0PtobethelineclosesttopwhichboundsahalfplaneinPcontainingallof!\PandwhosenormalinPformsanangleoflessthan=4withalltherayscontainedin!\P.LetLnPbetheclosestlinetopparalleltoL0andboundingahalfplaneinPcontaining n(Wjnin)\P.ThesetfL0;L1;L2;:::gisaninnitesequenceofparallellinesinPconstructedsothat%(p;Ln+1)6%(p;Ln).Toarriveatacontradictionitisenough13 toshowthat(%(p;Ln)�%(p;Ln+1))isboundedfrombelow.Nowthereexistsan"�0suchthatforanyx2Xthe"-ballcenteredatx,B(x;"),iscon-tainedinX\h1(X)\h�11(X)\h2(X)\h�12(X).Thereforewegetthat(%(p;L0)�%(p;L1))�".Nowweconsiderthecasewhen nis-hyperbolicforpositiveintegern.Foreveryy2 �1n(Ln),B(y;")iscontainedinthecomplementofhjnin(Wjn+1in+1).Andsincethean-glebetweenP\SnandthenormaltoLninPwasconstructedtobelessthan=4,B(x;"=23=2)forallx2Lniscontainedinthecomplementof n+1(Wjn+1in+1)and(%(p;Ln)�%(p;Ln+1))�"=23=2.Nowif nisnot-hyperbolicthenbyatheoremfromanotherpaperbyDrummandGoldmanwegetthat n+1=2is-hyperbolic.Wealsonoticethattheactionofg�1kdoesnotcontractanyvectorbymorethanafactorof(gk).Combiningthesefactswegetthat(%(p;Ln)�%(p;Ln+1))�(gk)"=23=2.Thereforewehavethat(%(p;Ln)�%(p;Ln+1))isboundedfrombelow.Sowegetacontradiction.Thus,thereisnop2R2;1whichisnotH-equivalenttoanelementofX.Hence,XisafundamentaldomainfortheactionofHonR2;1. 14 Chapter5TheGeometryofCrookedPlanesWesawinthelastchapterthatthefundamentaldomainsthatweconstructedforcertainMargulisspacetimes,areboundedbycertainpolyhedralhyper-surfacesinR2;1.Thesepolyhedralhyper-surfacesarecalledtheCrookedplanes.Acrookedplaneconsistsofthreeparts:twohalfplanes,calledwingsandapairofoppositeplanarsectors,calleditsstem.Thewingslieinnullplanesandthestem(whoseinteriorhastwoconnectedcomponents)liesinatimelike(indenite)plane. Inthischapterwestudytheintersectionsoftwocrookedplanes.5.1AnatomyofaCrookedPlaneLetp2R2;1beapointandv2R2;1aspacelikevector.DenethepositivelyorientedcrookedplaneC(v;p)R2;1withvertexpanddirectionvectorvtobetheunionoftwowingsW+(v;p):=p+P+(x+(v)),W�(v;p):=p+P+(x�(v))andastemS(v;p):=p+fx2R2;1:B(v;x)=0;B(x;x)60g.15 Eachwingisahalfplaneandthestemistheunionoftwoquadrantsinaspacelikeplane.Thepositivelyorientedcrookedplaneitselfisapiecewiselinearsubmanifold,whichstratiesintofourconnectedopensubsetsofplanes,fournullraysandavertex.NotethatforacrookedplaneC(v;p)withv2R2;1avectorandp2R2;1apointwecallthelinep+Rv,thespineofthegivencrookedplane.WealsodenethenegativelyorientedcrookedplaneK(v;p)R2;1withvertexpanddirectionvectorvtobetheunionoftwowingsM+(v;p):=p+P�(x+(v)),M�(v;p):=p+P�(x�(v))andastemS(v;p):=p+fx2R2;1:B(v;x)=0;B(x;x)60g.Inthefollowingsectionswewillonlyconsiderpositivelyorientedhalfplanes.Furthermore,anunspeciedorientationforacrookedplanewillassumedtobethepositiveone.Thecaseforthenegativelyorientedhalfplanesissimilar.Thefollowingsectionsdescribeintersectionsoftwowings,awingandastem,andtwostems.Fromtheseresultsfollownecessaryandsucientconditionsfortheintersectionoftwocrookedplanes.5.2IntersectionofWingsAplaneinR2;1maybewrittenasp+P(v),forp2R2;1andv2R2;1.Supposethatp1;p22R2;1andthatv1;v22R2;1arelinearlyindependent.ThentheintersectionP1\P2ofthetwoplanesPi:=pi+P(vi)isalinewhichcanbeparametrizedasp+R(v1v2)forsomep2P1\P2.Lemma5.2.1.Letx1;x22C+betwolinearlyindependentnullvectorsandp1;p22R2;1betwopoints.ThecorrespondingnullhalfplanesP+i:=pi+P+(xi)fori2f1;2garedisjointifandonlyifB(p2�p1;x1x2)�0.Otherwise,P+1\P+2isapointifandonlyifB(p2�p1;x1x2)=0andP+1\P+2isaspacelikelinesegmentifandonlyifB(p2�p1;x1x2)0.Proof.LetlbetheintersectionoftheplanesthatcontainP+1andP+2.ThenP+1\P+2l.Letw:=x1x2andp=l\(p1+P(w))sothatl=p+Rw.Thesubsetsl\P+ioflarecharacterizedasthesetofallp+kwwherek2RsatisfyingtheinequalitiesB((p+kw)�p1;�w)&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;0andB((p+kw)�p2;w)&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;0respectively.NowpwaschosensothatB(p�p1;w)=0andforthegivenparametrizationofll\P+1 !(�1;0]andl\P+2 ![B(p2�p1;w)=B(w;w);1).16 ThenP+1\P+2isemptyifandonlyifB(p2�p1;w)�0P+1\P+2isapointifandonlyifB(p2�p1;w)=0andP+1\P+2isaspacelikelinesegmentifandonlyifB(p2�p1;w)0.andtheproofiscomplete. Hereisthecorrespondinglemmaforpairsofhalfplaneswithoppositeorientations.Lemma5.2.2.Letx1;x22C+betwolinearlyindependentnullvectorsandp1;p22R2;1betwopoints.TheintersectionofthehalfplanesP+:=p1+P+(x1)andP�:=p2+P�(x2)isneverempty.Proof.Usingasimilarmethodofargumentasusedintheprevioustheoremwegettheresult. Nowweconsidertheintersectionofnullhalfplanesforthedegeneratecasewhenthenullhalfplanesareparallel:Lemma5.2.3.Letx2C+beanullvectorsandp1;p22R2;1betwopoints.ThecorrespondingnullhalfplanesP+i:=pi+P+(x)fori2f1;2garedisjointifandonlyifB(p2�p1;x)6=0.Otherwise,oneofthehalfplanescontainsthesimilarlyorientedhalfplaneandtheirintersectionisanullhalfplane.Proof.Usingasimilarmethodofargumentasusedinthersttheoremofthissectionwegettheresult. 5.3IntersectionofStemwithWingInthissectionwedescribewhenapositivelyorientednullhalfplaneintersectsastem.Fromthissectiononwardswewilljuststatethelemmasastheprooffollowsasimilarlineofreasoningaswasusedtoprovethersttheoremoftheprevioussection.17 Lemma5.3.1.LetxbeapositivelyorientednullvectorandvaunitspacelikevectorsuchthatB(x;v)0.Letp1;p22R2;1bepoints.ThenthestemS1:=S(v;p1)andthepositivelyorientednullhalfplaneP+2:=p2+P+(x)aredisjointifandonlyifB(p2�p1;vx)&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;jB(p2�p1;x)j.OtherwiseS1\P+2consistsof,apoint(whichlieson@S1butnoton@P+2)ifandonlyifB(p2�p1;vx)0=B(p2�p1;x).apoint(whichlieson@S1andon@P+2)ifandonlyifB(p2�p1;vx)=jB(p2�p1;x)j.aspacelikelinesegment(withatleastonevertexon@S1)ifandonlyifB(p2�p1;vx)jB(p2�p1;x)jwheretheotherendpointlieson@P+2ifandonlyifB(p2�p1;vx)&#x]TJ/;ཅ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;�jB(p2�p1;x)jandon@S1ifandonlyifB(p2�p1;vx)6�jB(p2�p1;x)j. Nowweconsiderthecasewhenthenullhalfplaneandthestemareorthogonal:Lemma5.3.2.LetxbeapositivelyorientednullvectorandvaunitspacelikevectorsuchthatB(x;v)=0.Letp1;p22R2;1bepoints.ThenthestemS1:=S(v;p1)andthepositivelyorientednullhalfplaneP+2:=p2+P+(x)aredisjointifandonlyifB(p2�p1;v)0whenxisamultipleofx�(v)B(p2�p1;v)&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;0whenxisamultipleofx+(v)OtherwiseS1\P+2isaraywithendpointon@P+2(exceptifB(p2�p1;x)=0,inwhichcaseitisanentireline).18 5.4IntersectionofStemsLetC(v1;p1)andC(v2;p2)betwocrookedplaneswherep1;p22R2;1arepointsandv1;v22R2;1areunitspacelikevectors.Wecallthetwovectorsv1andv2:ultraparalleli P(v1)\P(v2)isaspacelikelineorequivalentlyB(v1v2;v1v2)�0.asymptotici P(v1)\P(v2)isanulllineorequivalentlyB(v1v2;v1v2)=0.crossingi P(v1)\P(v2)isatimelikelineorequivalentlyB(v1v2;v1v2)0.Wedescribetheintersectionoftwostemsseparatelyfortheabovementionedthreedif-ferentcases.Lemma5.4.1.Letv1andv2beultraparallelunitspacelikevectors.ThestemsS(v1;p1)andS(v2;p2)aredisjointifandonlyifjB(p2�p1;v1v2)j&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;jB(p2�p1;v1)j+jB(p2�p1;v2)j.OtherwiseS(v1;p1)\S(v2;p2)isalinesegmentwhoseendpointslieontwodistinctlinescontainedin@S(v1;p1)[@S(v2;p2).Lemma5.4.2.Letv1andv2beasymptoticunitspacelikevectorssuchthatx�(v1)=x+(v2).ThestemsS(v1;p1)andS(v2;p2)aredisjointifandonlyifB(p2�p1;x�(v2)x+(v1))hasadi erentsignfromB(p2�p1;v1)andB(p2�p1;v2).OtherwiseS(v1;p1)\S(v2;p2)isapointifandonlyifB(p2�p1;x�(v2)x+(v1))=0andB(p2�p1;v1)andB(p2�p1;v2)havethesamesigns.alinesegmentifandonlyifB(p2�p1;x�(v2)x+(v1))andB(p2�p1;v1)andB(p2�p1;v2)havethesamesigns.arayifandonlyifeitherB(p2�p1;vi)=0foronlyoneiorB(p2�p1;v1)andB(p2�p1;v2)haveoppositesigns.alineifandonlyifB(p2�p1;vi)=0forbothi.Lemma5.4.3.Letv1andv2becrossingunitspacelikevectors.TheintersectionofthestemsS(v1;p1)andS(v2;p2)liesonalineandiseitheraline,theunionoftwodisjointrays,ortworaysandalinesegment,whereallendpointslieontheset@S(v1;p1)[@S(v2;p2).5.5IntersectionofCrookedPlanesInthissectionwedescribetheintersectionoftwocrookedplanes.Whileconsideringtheintersectionoftwocrookedplanes,someterminologyregardingtheorientationofthevectorsdeningthecrookedplanesisuseful.Saythatspacelikevectorsv1;v2;:::vnareconsistentlyorientedifB(vi;vj)0andB(vi;x(vj))6019 fori6=j.Geometrically,consistentorientationmeansthefollowing:AunitspacelikevectorvdenesahalfplaneH(v)inthehyperbolicplaneH2asfollows:H(v):=fu2U+:B(u;v)�0g.Supposev1;v22R2;1areultraparallel(respectivelyasymptotic)unitspacelikevectors.ThenthehalfplanesH(v1);H(v2)areboundedbyultraparallel(respectivelyasymp-totic)geodesicsinH2.Apairv1;v22R2;1ofultraparallel(respectivelyasymptotic)unitspacelikevectorsareconsistentlyorientedifandonlyifthehalfplanesH(v1);H(v2)aredisjoint.InthatcaseH(v1)\H(v2)isastripwithtwoidealboundarycomponentsandboundarycomponents@H(v1);@H(v2).Lemma5.5.1.Ifv1;v22R2;1aretwoconsistentlyorientedunitspacelikevectors,thenx+(v1)x+(v2)isapositivescalarmultipleof(v1v2)�v1+v2.x+(v1)x�(v2)isapositivescalarmultipleof(v1v2)+v1+v2.x�(v1)x+(v2)isapositivescalarmultipleof(v1v2)�v1�v2.x�(v1)x�(v2)isapositivescalarmultipleof(v1v2)+v1�v2.Proof.WeknowthatG0actstransitivelyonthesetofLorentzunitvectorssowithoutlossofgeneralitywecantakev1=2410035andx(v1)=1=p22401135:Applyingahyperbolicelementwithv1asaxedeigenvectorweget,v1=24�a0c35wherejcj=p a2�1.Furthermore,x(v2)=1=p224�c=a1=a135:Sincev1;v2areconsistentlyoriented,a;c�0.Thusa�1andc=p a2�1.Andforj;k2f�;+gwehavev1v2�kv1+jv2=24�k�ja�cjc35xj(v1)xk(v2)=24�j�k=a�c=ajc=a3520 xj(v1)xk(v2)isamultipleofv1v2�kv1+jv2,sinceB(xj(v1);v1v2v1+jv2)=0,B(xk(v2);v1v2�kv1v2)=0.Furthermore,xj(v1)xk(v2)isapositivemultipleofv1v2�kv1+jv2sinceB(xj(v1)xk(v2);v1v2�kv1+jv2)=(a1)2=a�0.Hencewehaveourdesiredresult. Theorem5.5.2.Letv1;v22R2;1betwoconsistentlyorientedultraparallelunitspacelikevectorsandp1;p22R2;1betwopoints.ThepositivelyorientedcrookedplanesC(v1;p1)andC(v2;p2)aredisjointifandonlyifjB(p2�p1;v1v2)j�jB(p2�p1;v1)j+jB(p2�p1;v2)j.Otherwisetheintersectioniseitherasinglepointorapolygon.Proof.Theresultfollowsfromthelemma5.4.1. Theorem5.5.3.Letv1;v22R2;1betwoconsistentlyorientedasymptoticunitspacelikevectorssuchthatx�(v1)=x+(v2)andp1;p22R2;1betwopoints.ThepositivelyorientedcrookedplanesC(v1;p1)andC(v2;p2)aredisjointifandonlyifB(p2�p1;v1)0,B(p2�p1;v2)0,B(p2�p1;x�(v2)x+(v1))&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;.51; 0 ;&#xTd [;0.Proof.Theresultfollowsfromapplyingthelemma5.4.2. Theorem5.5.4.Letv1andv2betwocrossingunitspacelikevectors.Theintersectionofcrookedplanesofindeterminateorientationwithspinesparalleltov1andv2isalwaysnonempty.Proof.Inthiscase,sincethestemsintersect,theorientationsareirrelevantandtheprooffollowseasilyfromthelemma5.4.3. Nowweconsidertheintersectionoftwocrookedplanesofdi erentorientations.Theorem5.5.5.Letv1andv2betwounitspacelikevectorswhicharenotparallel.TheintersectionofthepositivelyorientedcrookedplaneC(v1;p1)andthenegativelyorientedcrookedplaneK(v2;p2)isalwaysnonempty.Proof.Theresultfollowseasilyfromthelemmasintheprevioussections. 21 Chapter6TheExtendedMargulisInvariantandItsApplicationsInthischapterwetakeanalternateapproachtoattacktheoriginalproblem.WedescribeanextensionoftheMargulisinvarianttoacontinuousfunctiononhigherdimensionsgivenbyLabourieandfollowingtherecentworksofMargulis,Goldmanandlabourie,usetheextendedMargulisinvarianttogiveanequivalentcriterionforproperness.6.1FlatBundlesassociatedtoAneDeformationsInthissectionwemakethenecessarygroundworktoextendtheMargulisinvariant.LetusconsidertheLiegroupG0,thevectorspaceV=Vrandalinearrepresentation,L:G0�!GL(V).LetGbethecorrespondingsemi-directproductVoG0.MultiplicationinGisdenedby(v1;g1)(v2;g2):=(v1+g1v2;g1g2).Theprojection:G�!G0givenby(v;g)7�!gdenesatrivialbundlewithberVoverG0.Wenotethatwhenr=1,thatis,V=R2;1,GisthetangentbundleofG0withitsnaturalLiegroupstructure.SincetheberofequalsthevectorspaceV,thebrationcanbegiventhestructureofa(trivial)anebundleoverG0.Futhermore,thisstructureisG-invariant.DenotethetotalspaceofthisG-homogeneousanebundleoverG0by~E.Notethatwecanalsoconsiderasa(trivial)vectorbundle.ThisstructureisthenG0-invariant.ViaL,thisG0-homogeneousvectorbundlebecomesaG-homogeneousvectorbundle~VoverG0.Thisvectorbundleunderlies~E.TheG-homogeneousanebundle~EandtheG-homogeneousvectorbundle~Vadmit atconnections.Wedenotebothoftheseconnectionsby~r.WehaveseeninchaptertwothatLpreservesabilinearformBonV.TheG0invariantbilinearformB:VV�!Rdenesabilinearpairing22 B:~V~V�!Rofvectorbundles.Wenotethat,ifLpreservesabilinearformBonV,thebilinearpairingBon~Visparallelwithrespectto~r.Nowleta(t):=241000cosh(t)sinh(t)0sinh(t)cosh(t)35fort2R.Wenoticethatforp2H2,a(t)pdescribesthegeodesicthroughpandtangentto_p.Rightmultiplicationbya(�t)onG0identieswiththegeodesic ow~'tonUH2whereUH2denotestheunittangentbundleonH2.Wedenotethevectoreldcorrespondingtothegeodesic owby~'.Similarly,rightmultiplicationbya(�t)onGidentieswiththegeodesic ow~ton~E.Wenotethatthis owcoversthe ow'tonG0denedbyrightmultiplicationbya(�t)onG0.Alsothevectoreld~on~Egenerating~tcoversthevectoreld~'generating~'t.Wenotethatthe ow~tcommuteswiththeactionofG.Thus~tisa owonthe atG-homogeneousanebundle~Ecovering't.Wealsonotethatidentifying~Vasthevectorbundleunderlying~E,theR-actionisjustthelinearizationD~toftheaction~t.TheG-actionandthe owD~ton~Vpreserveasection~ofthebundle~V.Thissectioniscalledtheneutralsection.Wenotethatalthough~isnotparallelineverydirection,itisparallelalongthe ow~'t.Let�Gbeananedeformationofadiscretesubgroup�0G0.WedenotethequotientmanifoldH2=�0by.Since�isadiscretesubgroupofG,thequotientE:=~E=�isananebundleoverU=UH2=�0andinheritsa atconnectionrfromthe atconnection~ron~E.Furthermore,the ow~ton~Edescendstoa owtonEwhichisthehorizontalliftofthe ow'tonU.ThevectorbundleVunderlyingEisthequotientV:=~V=�=~V=�0andinheritsa atlinearconnectionrfromthe atlinearconnection~ron~V.The owD~ton~Vcovering~'tandtheneutralsection~bothdescendtoa owDtandasectionrespectively.WedenoteUrecUtobetheunionofallrecurrentorbitsof'.LetusdeneageodesiccurrentasaBorelprobabilitymeasureontheunittangentbundleUofinvariantunderthegeodesic ow't.WedenotethesetofallgeodesiccurrentsonbyC()andthesubsetofC()consistingofmeasuressupportedonperiodicorbitsbyCper().WenotethatC()hasthestructureofatopologicalspacewiththeweak?-topologyandhasanotionofconvexity.6.2Labourie'sdi usionoftheMargulisInvariantWerecallthattheMargulisinvariant = uisanRvaluedclassfunctionon�0whosevalueon 2�0equals23 B(u( )O�O;x0( ))whereOistheoriginandx0( )2Vistheneutralvectorof .NowtheoriginOwillbereplacedbyasectionsofE,theneutralvectorwillbereplacedbytheneutralsectionofV,andthelinearactionof�0onVwillbereplacedbythegeodesic owonU.LetsbeaC1sectionofE.Itscovariantderivativewithrespectto'isasmoothsectionr'(s).Andletbethenullsection.Wedeneafunction,Fu;s:U�!RwhereFu;s:=B(r'(s);).LetS(E)denotethespaceofcontinuoussectionssofEoverUrecwhicharedi er-entiablealong'andthecovariantderivativer'(s)iscontinuous.Ifs2S(E),thenFu;siscontinuous.Nowletusdene, [u];s():=RUFu;sd.Wenotethat [u];s()isindependentofthesectionswhichwasusedtodeneit.Sowegetawelldenedfunction, [u]:C()�!Rgivenby [u]()= [u];s()wheresisasection.Wealsonotethat [u]iscontinuousintheweak?-topologyonC().ThefollowingtheoremestablishesthelinkbetweenthisinvariantandtheMargulisinvariant.Itshowsthat [u]isindeedanextensionoftheMargulisinvariant.Theorem6.2.1.Let 2�0behyperbolicandlet 2Cper()bethecorrespondinggeodesiccurrent.Then ( )=`( )RUFu;sdwhere`( )isthelengthoftheclosedgeodesiccorrespondingtotheelement 2�0.6.3AnequivalentconditionforPropernessInthissectionwegiveanequivalentconditionforpropernessusingtheextendedMargulisinvariantandweconcludethissectionbygivingaconceptualproofoftheoppositesignlemma.Theorem6.3.1.Let�udenoteananedeformationof�0.Then�uactsproperlyifandonlyif [u]()6=0forall2C().NowusingthistheoremwegiveaproofoftheOppositesignlemma.Corollary6.3.2(OppositeSignLemma).Let 1; 22�0besuchthat ( 1)and ( 2)hasoppositesign.Then�doesnotactproperly.24 Proof.As ( 1)and ( 2)hasoppositesign,wecanwithoutlossofgeneralityassumethat ( 1)0 ( 2).Usingtheorem6:2:1weget [u]( 1)0 [u]( 2).NowconvexityofC()impliesthatthereexistsacontinuouspatht2C()witht2[1;2],forwhich1= 1and2= 2.Weknowthatthefunction [u]:C()�!Riscontinuous.Thereforetheintermediatevaluetheoremimpliesthat [u](t)=0forsomet2(1;2).Nowtheorem6:3:1impliesthat�doesnotactproperly. 25 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